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The book provides a new functional-analytic approach to evolution equations by considering the abstract Cauchy problem in a scale of Banach spaces. The usual functional analytic methods for studying evolution equations are formu lated within the setting of unbounded, closed operators in one Banach space. This setting is not adapted very well to the study of many pseudo differential and differential equations because these operators are naturally not given as closed, unbounded operators in one Banach space but as continuous opera tors in a scale of function spaces. Thus, applications within the setting of unbounded, closed operators require a considerable amount of additional work because one has to construct suitable closed realizations of these operators. This choice of closed realizations is technically complicated even for simple applications. The main feature of the new functional analytic approach of the book is to study the operators in scales of Banach spaces that are constructed by simple reference operators. This is a natural setting for many operators acting in scales of function spaces. The operators are only expected to respect the scale and to satisfy certain inequalities but we can avoid completely the choice of any closed realization of these operators which is of great importance in applications. We use the mapping properties of the reference operators to prove sufficient conditions for well-posedness of linear and quasilinear Cauchy problems. In the linear, time-dependent case these conditions are shown to characterize well-posedness. A similar result in the standard setting (i. e.
Inhaltsverzeichnis
1 Tools from functional analysis.- 1.1 A brief introduction into the theory of semigroups.- 1.2 Selfadjoint operators.- 1.3 Generators of analytic semigroups and their powers.- 1.4 Fractional Powers of operators of positive type.- 1.5 Complex interpolation spaces.- 1.6 Time-dependent, linear evolution equations.- 2 Well-posedness of the time-dependent linear Cauchy problem.- 2.1 Properties of well-posed linear Cauchy problems in scales of Banach spaces.- 2.2 Scales of Banach spaces generated by families of closed operators.- 2.3 Commutator estimates and scales of Banach spaces.- 2.4 Characterization of well-posedness of the Cauchy problem...- 2.5 Sufficient conditions for well-posedness of the Cauchy problem.- 3 Quasilinear Evolution Equations.- 3.1 Semilinear Evolution Equations.- 3.2 Commutator estimates and quasilinear evolution equations.- 3.3 A local existence and uniqueness result for quasilinear evolution equations.- 3.4 Regularity for quasilinear evolution equations in scales of Banach spaces.- 4 Applications to linear, time-dependent evolution equations.- 4.1 Pseudodifferential operators and weighted Sobolev spaces.- 4.2 Pseudodifferential evolution equations in scales of weighted Sobolev spaces.- 4.3 Essential selfadjointness of pseudodifferential operators.- 4.4 Evolution equations in C0(IRn) and Feller semigroups.- 4.5 Evolution equations in scales of Lq-Sobolev spaces.- 4.6 An application to a degenerate-elliptic boundary value problem.- 4.7 Evolution equations on networks.- 5 Applications to quasilinear evolution equations.- 5.1 Estimates of Nash-Moser type for differential operators.- 5.2 Quasilinear evolution equations in Sobolev spaces.- 5.3 Degenerate Navier-Stokes equations.- 5.4 The generalized Kadomtsev-Petviashvili equation.- 5.5 Quasilinear evolution equations in scales of Lq-Sobolev spaces.- 5.6 First order hyperbolic evolution equations in the C0k-scale.
Über den Autor / die Autorin
Dr. Oliver Caps, Universität Mainz
Zusammenfassung
The book provides a new functional-analytic approach to evolution equations by considering the abstract Cauchy problem in a scale of Banach spaces. Conditions are proved characterizing well-posedness of the linear, time-dependent Cauchy problem in scales of Banach spaces and implying local existence, uniqueness, and regularity of solutions of the quasilinear Cauchy problem.
Vorwort
Neuer funktional-analytischer Zugang
